This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369891 #11 Feb 04 2024 18:26:28 %S A369891 0,0,4,4,16,13,8,8,0,0,0,0,0,0,0,0,0,0 %N A369891 Minimum possible uncovered area when at most k squares of side k, k = 1..n, are packed into a square of side n*(n+1)/2 = A000217(n). %C A369891 The total area of the small squares is equal to the area of the large square, because 1^3+2^3+...+n^3 = (1+2+...+n)^2 = (n*(n+1)/2)^2 (Nicomachus's theorem). %C A369891 The small squares are assumed to be oriented in the same way as the large square. %C A369891 The partridge puzzle for size n is to determine whether a(n) = 0. The generalization considered here was suggested by _Rodolfo Kurchan_. %C A369891 Apparently, Robert T. Wainwright showed that a(12) = 0. This was also shown by Ågren et al., and Carl F. Schwenke noted that their solution can be modified to show that also a(14) = 0 and a(16) = 0. %C A369891 Is a(n) = 0 for all n >= 8? %H A369891 Magnus Ågren, Nicolas Beldiceanu, Mats Carlsson, Mohamed Sbihi, Charlotte Truchet, and Stéphane Zampelli, <a href="https://doi.org/10.1007/978-3-642-01929-6_3">Six ways of integrating symmetries within non-overlapping constraints</a>, in: Willem-Jan van Hoeve and John N. Hooker (eds), Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, CPAIOR 2009, Lecture Notes in Computer Science, vol 5547, Springer 2009; <a href="https://www.diva-portal.org/smash/record.jsf?pid=diva2:1042511">alternative link</a>. %H A369891 Pontus von Brömssen, <a href="/A369891/a369891.png">A perfect packing for n = 10, showing that a(10) = 0</a>. %H A369891 Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/0802.html">Problem of the Month (August 2002)</a>. %H A369891 Rodolfo Kurchan, <a href="https://www.puzzlefun.online/problems">Puzzle Fun</a> (see Partridge Puzzle). %H A369891 Robert T. Wainwright, <a href="https://mathpuzzle.com/partridge.html">The Partridge Puzzle</a>. %H A369891 Wikipedia, <a href="https://en.wikipedia.org/wiki/Squared_triangular_number">Squared triangular number</a>. %F A369891 a(2*k+1) <= a(2*k), because 2*k+1 squares of side 2*k+1 can be added in an L-shape to a square of side k*(2*k+1) to obtain a square of side (2*k+1)*(k+1). %Y A369891 Cf. A000217. %K A369891 nonn,more %O A369891 0,3 %A A369891 _Pontus von Brömssen_, Feb 04 2024